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education/math/MATH1210 (calc 1)/Derivatives.md
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@ -67,8 +67,8 @@ $$ \lim_{h \to 0} nx^{n-1} + P_{n3} x^{n-2}*0 \cdots v * 0 $$
The zeros leave us with: The zeros leave us with:
$$ f(x) = n, \space f'(x) = nx^{n-1} $$ $$ f(x) = n, \space f'(x) = nx^{n-1} $$
# Addition/Subtraction Derivative Rule # Sum and Difference Rules
You can add and subtract derivatives to find what the derivative of the whole derivative would be. $$ \dfrac{d}{dx}(f(x) \pm g(x)) = f'(x) \pm g'(x) $$
# Product Rule # Product Rule
$$ \dfrac{d}{dx} (f(x) * g(x)) = \lim_{h \to 0} \dfrac{f(x +h) * g(x + h) - f(x)g(x)}{h} $$ $$ \dfrac{d}{dx} (f(x) * g(x)) = \lim_{h \to 0} \dfrac{f(x +h) * g(x + h) - f(x)g(x)}{h} $$
@ -81,4 +81,6 @@ Then break into two different fractions:
$$\lim_{h \to 0} \dfrac{f(x + h)}{1} * \dfrac{(g(x + h) - g(x))}{h)} + \dfrac{g(x)}{1} *\dfrac{f(x + h) - f(x)}{h} $$ $$\lim_{h \to 0} \dfrac{f(x + h)}{1} * \dfrac{(g(x + h) - g(x))}{h)} + \dfrac{g(x)}{1} *\dfrac{f(x + h) - f(x)}{h} $$
From here, you can take the limit of each fraction, therefore showing that to find the derivative of two values multiplied together, you can use the formula: From here, you can take the limit of each fraction, therefore showing that to find the derivative of two values multiplied together, you can use the formula:
$$ \dfrac{d}{dx}(f(x) * g(x)) = f(x) * g'(x) + f'(x)* $$ \dfrac{d}{dx}(f(x) * g(x)) = f(x) * g'(x) + f'(x) $$
# Natural Exponential Function
$$ \dfrac{d}{dx} e^x = e^x $$